Wilkie’s conjecture for Pfaffian structures

We prove an effective form of Wilkie’s conjecture in the structure generated by restricted sub-Pfaffian functions: the number of rational points of height H lying in the transcendental part of such a set grows no faster than some power of log H. Our bounds depend only on the Pfaffian complexity of the sets involved. As a corollary we deduce Wilkie’s original conjecture for Rexp in full generality.